Fundamental class - Definition and Overview

In mathematics, the fundamental class is a homology class [M] associated to a manifold M. It is defined (firstly) in cases when M is a closed manifold of dimension n, and oriented. It is then an element of H_n(M,Z). If M is connected, that group is infinite cyclic, and it is the generator picked out by the given orientation.

It represents, in a sense, integration over M, and in relation with de Rham cohomology it is exactly that; namely for M a smooth manifold, an n-form ω can be paired with the fundamental class as

<ω, [M]>

to get a real number, which is the integral of ω over M, and depends only on the cohomology class of ω.